Solve:
1, Burnett explained Zhi Nuo's "dichotomy": that is, it is impossible to pass through an infinite number of points in a limited time, and you must pass through half of a given distance to complete the journey, so you must pass through half, and so on until infinity.
Aristotle criticized Zhi Nuo for making a mistake here: "He thinks that it is impossible for a thing to pass through infinite things in a limited time, or to contact infinite things alone. It should be noted that length and time are said to be "infinite" with two meanings.
Generally speaking, all continuous things are said to be "infinite", which has two meanings: either divided infinity or extended infinity. Therefore, on the one hand, things can't be connected with infinite things in a limited time.
On the other hand, it can contact with things that are divided into infinity, because time itself is divided into infinity. Therefore, things that pass through infinity are carried out in infinite time instead of finite time, and contact with infinite things is carried out in infinite times instead of finite times. ?
Aristotle pointed out that this argument is the same as the previous dichotomy. The conclusion of this argument is that people who run slowly can't catch up.
So the solution of this argument must be the same method. It is wrong to think that the leading thing in sports can't catch up, because it can't catch up in the leading time. However, if Zhi Nuo allows it to cross the prescribed limited distance, it can also be overtaken. ?
Aristotle thinks that Zhi Nuo's statement is wrong, because time is not composed of inseparable present, just as any other quantity is not composed of inseparable parts. Aristotle believes that this conclusion is caused by taking time as' now'. If we are not sure about this premise, this conclusion will not appear.
Aristotle thinks that the mistake here is that he regards the time when a moving object passes through another moving object as the time when it passes through a stationary object of the same size at the same speed, but in fact they are not equal.
Second, the rationality of calculus is seriously questioned, which almost subverts the whole calculus theory.
Solution: After Cauchy defined infinitesimal by the method of limit, the theory of calculus was developed and perfected, thus making the mathematics building more brilliant and beautiful!
Third, Russell Paradox: S is composed of all elements that do not belong to itself. Does s contain S? In layman's terms, one day Xiaoming said, "I'm lying!" "Ask xiao Ming is lying or telling the truth. The terrible thing about Russell's paradox is that it doesn't involve the profound knowledge of sets like the maximum ordinal paradox or the maximum cardinal paradox. It is simple, but it can easily destroy set theory!
solve
1. After the crisis, mathematicians put forward their own solutions. I hope to reform Cantor's set theory and eliminate the paradox by limiting the definition of set, which requires the establishment of new principles. "These principles must be narrow enough to ensure that all contradictions are eliminated; On the other hand, it must be broad enough so that all valuable contents in Cantor's set theory can be preserved. "
1908, Tzemero put forward the first axiomatic set theory system according to his own principles, which was later improved by other mathematicians and called ZF system. This axiomatic set theory system makes up for the defects of Cantor's naive set theory to a great extent. Besides ZF system, there are many axiomatic systems of set theory, such as NBG system proposed by Neumann et al.
2. Axiomatic set system successfully eliminates the paradox in set theory, thus successfully solving the third mathematical crisis. On the other hand, Russell's paradox has a far-reaching influence on mathematics. It puts the basic problems of mathematics in front of mathematicians for the first time with the most urgent needs, and guides mathematicians to study the basic problems of mathematics.
The further development of this aspect has profoundly affected the whole mathematics. For example, the debate on the basis of mathematics has formed three famous schools of mathematics in the history of modern mathematics, and the work of each school has promoted the great development of mathematics.
Extended data:
In the axiomatic system of class, some basic concepts are undefined, and we can only explain them in their objective sense, but such an explanation only helps to understand these concepts.
Any object studied in mathematics is called a class. The concept of class is infinite. There may be a relationship between classes called belonging. Class a belongs to class b, and class a is also called the element of class B.
We can understand a class as a whole composed of several elements. Whether a class is an element of another class is completely certain, which is the certainty of class elements. If class a is not an element of class b, it is said that a does not belong to B.
References:
Baidu Encyclopedia-The Third Mathematical Crisis
References:
Baidu Encyclopedia-The Second Mathematical Crisis
References:
Baidu Encyclopedia-The First Mathematical Crisis
References:
Baidu Encyclopedia-Three Crises of Mathematics